Stokes' Theorem // Geometric Intuition & Statement // Vector Calculus

We're finally at one of the core theorems of vector calculus: Stokes' Theorem. We've seen the 2D version of this theorem before when we studied Green's Theorem which compared the circulation around a 2D curve to integrating the circulation density along the region. In contrast, Stokes Theorem is the three-dimensional generational to compare the circulation of a 3D curve in some vector field to the integral over the region of the curl of the vector field (note: the kth component of curl is what we used to call the circulation density). In this video we build up the geometric conceptual understanding of why the curl of a vector field would relate to the line integral along it's boundary, and then finally state the theorem.
0:00 The Geometric Picture
3:30 Recalling Green's Theorem
5:55 Stating Stokes' Theorem
MY VECTOR CALCULUS PLAYLIST:
►VECTOR CALCULUS (Calc IV) • Calculus IV: Vector Ca...
OTHER COURSE PLAYLISTS:
►DISCRETE MATH: • Discrete Math (Full Co...
►LINEAR ALGEBRA: • Linear Algebra (Full C...
►CALCULUS I: • Calculus I (Limits, De...
► CALCULUS II: • Calculus II (Integrati...
►MULTIVARIABLE CALCULUS (Calc III): • Calculus III: Multivar...
►DIFFERENTIAL EQUATIONS: • How to solve ODEs with...
OTHER PLAYLISTS:
► Learning Math Series
• 5 Tips To Make Math Pr...
►Cool Math Series:
• Cool Math Series
BECOME A MEMBER:
►Join: / @drtrefor
MATH BOOKS & MERCH I LOVE:
► My Amazon Affiliate Shop: www.amazon.com/shop/treforbazett
SOCIALS:
►Twitter (math based): / treforbazett
►Instagram (photography based): / treforphotography

[@BenTzionZuckier]

I've been stoked for this

[@DrTrefor]

oh u:D

[@agrajyadav2951]

Heck yeah lol

[@mustafizurrahman5699]

Sir I had studied this topic in 1998 or 1999 but couldn't fathom anything. After 25 years, when I am 47 years old, I have been able to understand the intuition. Thank you awfully for enthralling explanation

[@pankajjkumar369]

Your geometric presentation is mind blowing.

[@naman4067]

You can plot this on maple calculator too

[@kieranvelasquez5670]

This is how every math lecture should be: 1.what this looks like/does 2.how to solve/do the problem 3.what this applies to Awesome video made this theorem tie together everything from multi variable calculus so well!

[@bensonjordan103]

Sorry to be off topic but does anybody know of a way to log back into an instagram account..? I was stupid lost my password. I would love any assistance you can give me

[@deanxavier9552]

@Benson Jordan Instablaster ;)

[@bensonjordan103]

@Dean Xavier i really appreciate your reply. I got to the site on google and im trying it out atm. Takes quite some time so I will reply here later with my results.

[@nitinreganti8033]

@@bensonjordan103 did it work!

[@agrajyadav2951]

Nah i think that a lecture that introduces an idea should start with a question, and build up to the theorem without exposing it first hand. That makes it much more interactive and helps us appreciate the beauty of the idea

[@ahmeds5181]

This channel deserves a million subscriberd!! Highly underrated channel..

[@kozukioden2406]

Forget about the maths, you need to teach me how to grow a beard like this

[@tasninnewaz6790]

Geometric Intuition gives me geometric feeling. Thank you Trefor!

[@chaithrapai4266]

Lots of love and respect from India.....♥️♥️♥️♥️

[@manojsahoo8303]

Have no words to praise u sir

[@NumbToons]

3:20 Words of wisdom. 'Tendency to curl withiiiiin the surface' HOOOLY MOTHER OF ACCURATE DESCRIPTION.

[@judepazier4491]

I have begun watching your videos and this all is beginning to make so much sense! Vector calculus can get really abstract/difficult to picture but you made it clear and precise. Thank you for sharing!

[@keyamoni1602]

Whenever i'm having problem with understanding vector calculus or others complicated maths i come here and my doubt gets cleared. This channel is like a lighthouse. Whenever i lose my way it helps me.

[@user-iu2il6np5y]

Dear Dr. mostly I am watching your lectures, in each lecture I realize that your geometrical representation of the problem is mind blowing and amazing.

[@briandwi2504]

Fantastic. Such a great play list. I have learned so much!

[@transcendingvictor]

Amazing and enlightening video, it is like if the strange relations the Stoke's and Green's theorems describe now made sense. Thank you very much!!

[@abstractnonsense3253]

You are well on your way to having the overall best set of video lectures on KZbin about the core of university mathematics. Discrete Mathematics, Linear Algebra, Calculus 1, and Calculus 2 are done. Calculus 3 is almost done. All there is left is Complex Analysis, Differential Equations, and Probability & Statistics. Really great work! Impressive to anyone who knows how challenging it is to teach mathematics!

[@DrTrefor]

Thank you! That....uh....is the plan:D Next up is differential equations:)

[@krupt5995]

@@DrTrefor You are awesome dude. And what amazes me is how you can keep a happy, energetic and passionate smile the whole time you are talking. Amazing teaching skills

[@sergiolucas38]

Amazing video as always, finally i got to these advanced concepts, so crucial to understand field fenomena in physics, thanks :)

[@Chomusuke71]

So lucky to get this in my first year itself...beautiful playlist sir!!

[@DrTrefor]

Thank you so much

[@khatri_17]

I think I finally found the channel I need to visualize my maths imagination.

[@woodychelton5590]

really like the visualization, great stuff

[@sumitkumarsahoo7601]

AWESOME, great explanation, doing a great job

[@dodo-js5gw]

Thanks a lot ! It is much more intuitive now ! Hello from France :D

[@raghuramabl6729]

Thanks a lot, you cleared my confusion regarding what exactly is going on when we apply curl on each point of the surface once again thank you 🙌👌🙏

[@sambuddharanjankhanra376]

I chose ur video over kan academy's video keep up the good work

[@spiderkent]

I never really understand the idea behind the theory until I watch this video. Thank you very much.

[@mmariokart231]

Ah! Thank you so much! That really helps tie it together

[@el_andrius]

Thanks for such rich experience!

[@AnkurSingh-cv1ls]

that's what i was looking for......perfect explanation and all these theorems amazed me......beautiful

[@DrTrefor]

Isn’t it so pretty?

[@AnkurSingh-cv1ls]

@@DrTrefor it is and so does your explanation......really loved it!!

[@qamarmoavia4031]

Thank U so much Dr . Keep going ........

[@user-wu8yq1rb9t]

I just watched and enjoyed! And for me, when I'm Watching your videos, it's just look like I'm watching "Big Bang theory" or "Friends" series. Thank you so much 💓

[@tendi.rutsate]

Love the way you explain. Thank you!

[@DrTrefor]

You are so welcome!

[@jlpsinde]

Love from Portugal! Realy top

[@majidalishabbeer]

Truly immaculate!

[@gregorsamsa9762]

Thank you so much!

[@shubhamg23]

Brilliant!! The beauty with which you connected the two thrms...🔥🔥🔥🔥🔥

[@DrTrefor]

Isn’t it cool!?

[@shubhamg23]

@@DrTrefor Certainly!!..The Instructor As Well👍🤩

[@sureshpatel-sg2xw]

@@DrTrefor its cool and fun too

[@dhanishcreations7756]

I love you. You are a life saver.

[@Chomusuke71]

This is pure gold!!

[@DrTrefor]

Thank you so much!

[@eslamelsofany2470]

thank you for these amazing videos

[@DrTrefor]

You’re most welcome!!

[@physicsdiva1871]

Nice explanation

[@Kottam_Yallawa]

Fluid kinematics is all about vector calculus and now i learnt both 🙂

[@vibudha_keshava]

I'm currently doing my PhD and deal with Stokes' theorem a lot. Particularly using partial integration and product rule on Stokes' theorem to regularize certain singular integrals in Boundary Element Method. Would love for some discussion and exchanges with you :)

[@football4d149]

So nice of you amazing video amazing feeling

[@DrTrefor]

Thank you so much!

[@RUDRARAKESHKUMARGOHIL]

if not understood or a absence of feeling is lacking at some point you may surely watch niltime animation of stokes theorem it will surely fill the loop hole....by the way a great explanation thankyou guruji...

[@amanravan9795]

Simply wow

[@robertoberidojr.435]

I love this video

[@lucaschis4529]

Beautiful 😎

[@testwest8878]

Hi Dr. Trefor, I recently received a question that required us to use Stoke's Theorem to find the flow rate of the gases escaping through the surface of a sphere with a hole at the bottom. However, it seems that Stoke's Theorem finds the circulation not flux. Why do we use Stoke's Theorem in this case?

[@christianzabala7839]

Sir. May I ask what software are you using? The one letting you draw on the graph

[@sudiptaroyarts3861]

I'm always telling that you are awesome sir ❤❤❤❤

[@DrTrefor]

I"m always appreciating it:)

[@bawol-official]

I don’t know if this is the place to ask but I’m about to take my Calc 3 final and I want to ask for ways that I can keep my skills sharp. Is there any mathematical software/calculus visualization sites that can keep me interested once I finish the class?

[@geoffreyfaust3443]

Another great video, Dr. B! FYI, we non-Canadians are jarred more by your pronunciation of "corollaries" than of "zed". Cheers! G :

[@user-iu2il6np5y]

Dear Dr. can you tell me the code how to plot the vector field of 2D or 3D vector in GeoGebra App

[@richardvalentinonainggolan328]

at 4:30, can someone please explain what is the k component of the curl? I saw the previous video but still don't understand about that part.

[@madhavkulkarni8980]

Dear Trefor From last ten years I am trying to understand idea behind green & stokes theorem after going through lot of books & websites I unable to capture the concept. So I am very much thankful. Secondly will you please explain cauchys theorem, cauchys integral formula & Riemann mapping theorm which I unable to understand how any deformed region can be transformed to unit circle? wishing all the best for the roject. Thanks again Regards MADHAV KULKARNI M. Sc. ( Maths)

[@j.o.5957]

So (dN/dx - dM/dy) * k_hat gives circulation with the normal in the k_hat direction. Then nabla X F gives all the (dN/dx - dM/dy) * k_hat + (...). dot it by n and you get the tendency to curl around the normal. Take all of that and integrate it over the surface and you get the circulation. Question to self: how to get the normal vectors in the opposite direction? Based on previous observations, I'm assuming we take dM/dy - dN/dx etc. by using the right hand rule.

[@Aaron-lp3zt]

Stokes' theorem is lowkey 🔥

[@DrTrefor]

It really is:D

[@Aaron-lp3zt]

@@DrTrefor He really did my boi Green dirty with that generalization tho

[@Jechew]

I’ve been wondering. What application do you use to graph out everything for your videos? I at one point thought it was manual modeling, but apparently not

[@djredrover]

Hi Dr. Bazett, do you have any plans on doing any videos on linear algebra, specially state space modelling of rockets dynamics and transition between different coordinates and quaternions? I would absolutely LOVE your take on these topics. Thanks again for you wonderful channel.

[@DrTrefor]

I do have a linear algebra playlist, but probably not something as specific as that

[@djredrover]

@@DrTrefor Woops. Totally missed that playlist. Can't wait to dig in. PS. The length of your videos are bang-on on the "detail vs. bored'm" graph, if you will.

[@miguelaphan58]

You SEE there are profesors..and Masters ..Bazzet for instance..is a master..a Big one

[@hesammozafarinia8301]

Hi Dr. Trefor, I recently received a question about Stoke's Theorem That is : why that circulation density is just the K(th) component of curl ? , iknow it can proved by the formula of the curl ... but as i understood the curl tells us the circulation of a point then why when we are in xy plane the net circulation is the K component of our curl ?

[@joaocosta3506]

That's because in the xy plane your rotation can only be about the K axis.

[@jaikumar848]

Hello sir ! Could you please tell the the practical application of this theorem?

[@DrTrefor]

Probably the biggest use is in elecricit and magnetism where stokes theorem and divergence theorem are HUGE

[@Chomusuke71]

maxwells equations

[@justadreamerforgood69]

Anything in electromagnetics or fluid mechanics

[@kakashi8203]

El Professor 🙏

[@abstractnonsense3253]

Reaching the mountain top! 👍

[@DrTrefor]

You know it!

[@TriThom50]

Why does the curl on the interior always cancel? Could we not imagine some vector field that is purely divergent with 0 curl? Similarly, what if the curl is more at certain points than others? How can we use the logic that it will cancel in that case?

[@danishmalik2295]

Ty

[@kesumanojkumar2074]

Hey need to contact you about subject any possibility

[@sukranochani5764]

Nice.

[@DrTrefor]

Thank you! Cheers!

[@krupt5995]

But there are infinite surface's that can 'support' the same boundary/curve. I suppose this is true for all possible surfaces?

[@user-qu5zm3yn3f]

good

[@jinks908]

*Sits in lecture*: I don’t get it. *Reads textbook*: I still don’t get it *Works through examples*: Still no. *Watches Dr. TB*: Oohhhhhhhh!!!!!

[@salimbenchekroun7543]

Is this how we model Eddy currents?

[@DrTrefor]

absolutely, all this vector calculus stuff is core to understanding fluid dynamics

[@AbhinavK-no4ij]

Sir why there is no dxdy instead of dsigma

[@dunus8076]

becuz we are generalizing to a surface. dxdy is a special case for when the surface is on the xy-plane completely.

[@bartholomeosphinx4382]

Imagine you're listening to a lecture where the lecturer jerks and jumps every 10 seconds.

[@brendan12882]

3:56 clounter what now? :O

[@Nobody-zq8bl]

"Stokes' " implies plural possessive of Stokes, more than one Stokes. It should be Stokes's, but thank you.

[@master_zenrade]

The one disliker must be insane.

[@DrTrefor]

Hahah agreed!

[@dkutlay1282]

you should have said "I'm STOKED for Stokes" :(

[@adiej4350]

dude i thought god said he wasnt sending any more prophets / angels down wyd here ????

[@ES-qe1nh]

🎉

[@seanlinehan8478]

Who asked?

[@ES-qe1nh]

Yuri